Somehow I'd never read this one, though it was a set text for some fellow-pupils at my school (can't remember if it was other classes at O-Level, or more likely those who did English at A-Level). In 1980s Belfast, the story of persecution of the Catholic church and its adherents by the agents of the State had local nuances which were not lost on any of us. Thirty years of scandal later, it's rather more difficult to see the church in potentially heroic light.
Of course Greene was a Catholic writer writing from an English point of view, and I wonder quite how true to Mexican religious practice his portrayal is - yes, I know that he had gone to Mexico for four months in 1938 to see the situation on the ground for himself, but it's also pretty clear that he went and returned with a narrative already in his mind. That side of things doesn't matter much now; Tomás Garrido Canabal, the Tabasco governor whose anti-clericalism Greene reported on, died in 1943, and the Catholic church has become its own worst enemy in Mexico as elsewhere.
Anyway, I think such a reading is far from the intended core of the book. Greene's real theme is heroism and redemption - an unlikely hero who finds it in himself to do the right thing, having been doing many of the wrong things, written at the outbreak of the Second World War when the Zeitgeist needed unlikely heroes. The unnamed hero has made a real mess of his life, and of other people's, but finds a moment or two when he can make a difference and rescue his own dignity. That much is a story that can be told in many times and places.